3.29.80 \(\int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2)} \, dx\)

Optimal. Leaf size=311 \[ -\frac {\log \left (a^2 d^{2/3}+\left (\sqrt [3]{d} x-a \sqrt [3]{d}\right ) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+\left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a \sqrt [3]{d}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{\sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 9.85, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-(a*b) - a*c + 2*b*c + (2*a - b - c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(-(b*c) + a^2*d + (b + c - 2*
a*d)*x + (-1 + d)*x^2)),x]

[Out]

((2*a - b - c + Sqrt[b^2 + c^2 + 4*a^2*d - 4*a*c*d - 2*b*(c + 2*a*d - 2*c*d)])*(-a + x)^(1/3)*(-b + x)^(1/3)*(
-c + x)^(1/3)*Defer[Int][1/((-a + x)^(1/3)*(-b + x)^(1/3)*(-c + x)^(1/3)*(b + c - 2*a*d - Sqrt[b^2 - 2*b*c + c
^2 + 4*a^2*d - 4*a*b*d - 4*a*c*d + 4*b*c*d] + 2*(-1 + d)*x)), x])/(-((a - x)*(b - x)*(c - x)))^(1/3) + ((2*a -
b - c - Sqrt[b^2 + c^2 + 4*a^2*d - 4*a*c*d - 2*b*(c + 2*a*d - 2*c*d)])*(-a + x)^(1/3)*(-b + x)^(1/3)*(-c + x)
^(1/3)*Defer[Int][1/((-a + x)^(1/3)*(-b + x)^(1/3)*(-c + x)^(1/3)*(b + c - 2*a*d + Sqrt[b^2 - 2*b*c + c^2 + 4*
a^2*d - 4*a*b*d - 4*a*c*d + 4*b*c*d] + 2*(-1 + d)*x)), x])/(-((a - x)*(b - x)*(c - x)))^(1/3)

Rubi steps

\begin {align*} \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \left (\frac {2 a-b-c+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )}+\frac {2 a-b-c-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\left (2 a-b-c-\sqrt {b^2+c^2+4 a^2 d-4 a c d-2 b (c+2 a d-2 c d)}\right ) \sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \frac {1}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (\left (2 a-b-c+\sqrt {b^2+c^2+4 a^2 d-4 a c d-2 b (c+2 a d-2 c d)}\right ) \sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \frac {1}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ \end {align*}

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Mathematica [F]  time = 9.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-(a*b) - a*c + 2*b*c + (2*a - b - c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(-(b*c) + a^2*d + (b +
c - 2*a*d)*x + (-1 + d)*x^2)),x]

[Out]

Integrate[(-(a*b) - a*c + 2*b*c + (2*a - b - c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(-(b*c) + a^2*d + (b +
c - 2*a*d)*x + (-1 + d)*x^2)), x]

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IntegrateAlgebraic [A]  time = 3.15, size = 311, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-(a*b) - a*c + 2*b*c + (2*a - b - c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(-(b*c) + a^2*
d + (b + c - 2*a*d)*x + (-1 + d)*x^2)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))/(-2*a*d^(1/3) + 2*d^
(1/3)*x + (-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))])/d^(1/3) + Log[a*d^(1/3) - d^(1/3)
*x + (-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3)]/d^(1/3) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*
x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3) +
(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(2/3)]/(2*d^(1/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*b-a*c+2*b*c+(2*a-b-c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x,
algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {a b + a c - 2 \, b c - {\left (2 \, a - b - c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*b-a*c+2*b*c+(2*a-b-c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x,
algorithm="giac")

[Out]

integrate(-(a*b + a*c - 2*b*c - (2*a - b - c)*x)/((-(a - x)*(b - x)*(c - x))^(1/3)*(a^2*d + (d - 1)*x^2 - b*c
- (2*a*d - b - c)*x)), x)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {-a b -a c +2 b c +\left (2 a -b -c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {1}{3}} \left (-b c +a^{2} d +\left (-2 a d +b +c \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*b-a*c+2*b*c+(2*a-b-c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x)

[Out]

int((-a*b-a*c+2*b*c+(2*a-b-c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {a b + a c - 2 \, b c - {\left (2 \, a - b - c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*b-a*c+2*b*c+(2*a-b-c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x,
algorithm="maxima")

[Out]

-integrate((a*b + a*c - 2*b*c - (2*a - b - c)*x)/((-(a - x)*(b - x)*(c - x))^(1/3)*(a^2*d + (d - 1)*x^2 - b*c
- (2*a*d - b - c)*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {a\,b+a\,c-2\,b\,c+x\,\left (b-2\,a+c\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{1/3}\,\left (x\,\left (b+c-2\,a\,d\right )-b\,c+a^2\,d+x^2\,\left (d-1\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*b + a*c - 2*b*c + x*(b - 2*a + c))/((-(a - x)*(b - x)*(c - x))^(1/3)*(x*(b + c - 2*a*d) - b*c + a^2*d
+ x^2*(d - 1))),x)

[Out]

int(-(a*b + a*c - 2*b*c + x*(b - 2*a + c))/((-(a - x)*(b - x)*(c - x))^(1/3)*(x*(b + c - 2*a*d) - b*c + a^2*d
+ x^2*(d - 1))), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*b-a*c+2*b*c+(2*a-b-c)*x)/((-a+x)*(-b+x)*(-c+x))**(1/3)/(-b*c+a**2*d+(-2*a*d+b+c)*x+(-1+d)*x**2),
x)

[Out]

Timed out

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